Mathematical clarification for step-44

[Arnold]

Hello,

I don't get the formula for Jc_vol in step-44 (marked purple), particulary the fourth order unit tensor in there. I looked up the source that is quoted in step-44 (Miehe 1994), and the formula for db^-1/db uses a unit tensor as well (see picture for details).

I'm not sure however if the formula Miehe uses is the correct one for _symmetric_ tensors. It should theoretically be equal to the formula below from "Nonlinear Solid Mechanics", but when I try to evaluate the Miehe formula in index notation I get a weird result that is much different from the Holzapfel formula.

Can someone confirm with certainty that these two formulas (marked red) are actually the same thing? Or even better, locate the error in my derivation attempt?

Best Regards,

Arnold

[Jean-Paul Pelteret]

Hi Arnold,

When I compare the documentation to step-44 that you quote to the code itself, it would seem like there’s a typo there. The second term “-2 p \mathcal{I}” of the part that underlined should actually read “-2 p \mathcal{S}” since this really is the symmetric fourth order identity tensor that holds the symmetry properties that you also quoted.

To get from the Holzapfel equation that you quoted to the one used in the tutorial, you still need to “push forward” these quantities. The tensor 6.166 is the volumetric tangent described in the reference configuration, while the tutorial / Miehe describe a tangent in the spatial configuration. The appropriate transformation is the Piola transformation. 

As for your derivation, I’m not quite sure that I follow what you’re trying to do here. The identity that is applied at the equation below Miehe (57) — let’s take it at face value that its true — is used to simplify the calculation in (58) where the formal definition of the tangent involves the contraction from both sides of the derivative of the energy function by b, the left Cauchy-Green tensor. Since the identity involves produces to b^{-1} on either side of the symmetric fourth order unit tensor, they cancel out in (58). To verify your result, you’ll need to compute the LHS of the equation, i.e. \partial_{b} b^{-1}. But, in fact, this entire identity comes from the statement

0 = d/db (I)    where I is the 2nd order identity tensor

   = d/db (b^{-1} . b)   where the “.” denotes a single contraction

   = [db^{-1}/db] . b + b^{-1} . [db/db]

  =  [db^{-1}/db] . b + b^{-1} . S     where S is the forth order unit identity tensor (this is a standard derivation, specialised for the case of b being symmetric)

—> contracting by b^{-1} on the right and rearranging —>

 db^{-1}/db = -  b^{-1} . S . b^{-1}

I must say that, for me at least, trying to get straight to the result as Miehe does is not easy to understand unless you know some rather unusual (maybe?) identities. I prefer the to start in the reference configuration, as Holzapfel does, and then push forward all results to the spatial configuration. That way, the individual step are, in my opinion, quite clear albeit lengthy. There are some notes in Appendix A.3 of my dissertation that sketch it out. 

I hope that helps you a little.

Best,

Jean-Paul

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